17.2. Principal connections. Consider a principal ¬ber bundle (P, p, M, G)

with the principal action r : P — G ’ P . We shall also denote by r the canonical

right action r : J 1 P — G ’ J 1 P given by rg (jx s) = jx (rg —¦ s) for all g ∈ G

1 1

and jx s ∈ J 1 P . In accordance with 11.1 we de¬ne a principal connection “ on

1

a principal ¬ber bundle P with a principal action r as an r-equivariant section

“ : P ’ J 1 P of the ¬rst jet prolongation J 1 P ’ P .

Let us recall that for every principal bundle, there are the canonical right

actions of the structure group on its tangent bundle and vertical tangent bundle.

By de¬nition, for every vector ¬eld ξ ∈ X(M ) and principal connection “ the “-

lift “ξ is a right invariant projectable vector ¬eld on P . Furthermore, a principal

connection induces an identi¬cation J 1 P ∼ V P — T — M which maps principal

=

connections into right invariant sections.

17.3. Induced connections on associated ¬ber bundles. Let us consider

an associated ¬ber bundle E = P [S; ]. Every local section σ of P determines a

local trivialization of E. Hence the idea of the de¬nition of induced connections

used in 11.8 gets the following simple form. For any principal connection “ on

P we de¬ne the section “E : E ’ J 1 E by “E {u, s} = jx {σ, s}, where u ∈ Px

1

ˆ

1

and s ∈ S are arbitrary, “(u) = jx σ and s means the constant map M ’ S

ˆ

with value s. It follows immediately that the parallel transport PtE (c, {u, s}) of

an element {u, s} ∈ E along a curve c : R ’ M is the curve t ’ {Pt(c, u, t), s}

where Pt is the G-equivariant parallel transport with respect to the principal

connection on P .

We recall the canonical principal bundle structure (T P, T p, T M, T G) on T P

and T E = T P [T S, T ], see 10.18. The horizontal lifting determined by the

induced connection “E is given for every ξ ∈ X(M ) by

“E ξ({u, s}) = {“ξ(u), 0s } ∈ (T E)ξ(p(u)) ,

(1)

where 0s ∈ Ts S is the zero tangent vector. Let us now consider an arbi-

trary general connection “E on E. Chosen an auxiliary principal connection

“P on P , we can express the horizontal lifting γE in the form “E ξ({u, s}) =

{“P ξ(u), γ (ξ(p(u)), s)}. The map γ is uniquely determined if the action is in-

¯ ¯

¬nitesimally e¬ective, i.e. the fundamental ¬eld mapping g ’ X(S) is injective.

Then it is not di¬cult to check that the horizontal lifting γE can be expressed

in the form (1) with certain principal connection “ on P if and only if the map

γ takes values in the fundamental ¬elds on S. This is equivalent to 11.9.

¯

17.4. The bundle of (principal) connections. We intend to treat principal

connections as sections of an appropriate bundle. We have de¬ned them as right

invariant sections of the ¬rst jet prolongation of principal bundles, so that given

a principal connection “ on (P, p, M, G) and a point x ∈ M , its value on the

whole ¬ber Px is determined by the value in any point from Px . We de¬ne QP

to be the set of orbits J 1 P/G. Since the source projection ± : J 1 P ’ M is G-

invariant, we have the projection QP ’ M , also denoted by ±. Furthermore, for

¯¯¯ ¯

every morphism of principal ¬ber bundles (•, •1 ) : (P, p, M, G) ’ (P , p, M , G)

¯

over •1 : G ’ G it holds

J 1 •(jx (ra —¦ s)) = j•0 (x) (r•1 (a) —¦ • —¦ s —¦ •’1 )

1 1

0

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

160 Chapter IV. Jets and natural bundles

¯

for all jx s ∈ J 1 P , a ∈ G. Hence the map J 1 • : J 1 P ’ J 1 P factors to a map

1

¯

Q• : QP ’ QP and Q becomes a functor with values in ¬bered sets. More

¯

explicitly, for every jx s in an orbit A ∈ QP the value Q•(A) is the orbit in J 1 P

1

going through J 1 •(jx s). By the construction, we have a bijective correspondence

1

between the sections of the ¬bered set QP ’ M and the G-equivariant sections

of J 1 P ’ P which are smooth along the individual ¬bers of P . It remains to

de¬ne a suitable smooth structure on QP .

Let us ¬rst assume P = Rm — G. Then there is a canonical representative

in each orbit J 1 (Rm — G)/G, namely jx s with s(x) = (x, e), e ∈ G being

1

the unit. Moreover, J 1 (Rm — G) is identi¬ed with Rm — J0 (Rm , G), jx s ’

1 1

(x, j0 (pr2 —¦ s —¦ tx )). Hence there is the induced smooth structure Q(Rm — G) ∼

1

=

m 1 m 1 m m

R — J0 (R , G)e and the canonical projection J (R — G) ’ Q(R — G)

becomes a surjective submersion. Let PBm be the category of principal ¬ber

bundles over m-manifolds and their morphisms covering local di¬eomorphisms

¯

on the base manifolds. For every PBm -morphism • : Rm — G ’ Rm — G and

element jx s ∈ A ∈ Q(Rm — G) with s(x) = (x, e), the orbit Q•(A) is determined

1

by J 1 •(jx s). This means that

1

’1

—¦ • —¦ s —¦ •’1 )

Q•(jx s) = j•0 (x) (ra

1 1

0

where a = pr2 —¦ •(x, e) and consequently Q• is smooth.

Now for every principal ¬ber bundle atlas (U± , •± ) on a principal ¬ber bundle

P the maps Q•± form a ¬ber bundle atlas (U± , Q•± ) on QP ’ M . Let us

summarize.

Proposition. The functor Q : PBm ’ FMm associates with each principal

¬ber bundle (P, p, M, G) the ¬ber bundle QP over the base M with standard

¬ber J0 (Rm , G)e . The smooth sections of QP are in bijection with the principle

1

connections on P .

The functor Q is a typical example of the so called gauge natural bundles

which will be studied in detail in chapter XII. On replacing the ¬rst jets by

k-jets in the above construction, we get the functor Qk : PBm ’ FMm of k-th

order (principal) connections.

17.5. The structure of an associated bundle on QP . Let us consider a

principal ¬ber bundle (P, p, M, G) and a local trivialization ψ : Rm — G ’ P .

By the de¬nition, the restriction of Qψ to the ¬ber S := (Q(Rm — G))0 is a

di¬eomorphism onto the ¬ber QPψ0 (0) . Since the functor Q is of order one, this

di¬eomorphism is determined by j 1 ψ(0, e) ∈ W 1 P , cf. 15.3. For the same reason,

every element j 1 •(0, e) ∈ Wm G determines a di¬eomorphism Q•|S : S ’ S. By

1

1

the de¬nition of the Lie group structure on Wm G, this de¬nes a left action of

Wm G on S. We de¬ne a mapping q : W 1 P — S ’ QP by

1

q(j 1 ψ(0, e), A) = Qψ(A).

Since q(j 1 (ψ —¦•)(0, e), Q•’1 (A)) = Qψ —¦Q•—¦Q•’1 (A), the map q identi¬es QP

with W 1 P [S; ]. We shall see in chapter XII that the map q is an analogy to our

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

17. Connections and the absolute di¬erentiation 161

identi¬cations of the values of bundle functors on Mfm with associated bundles

to frame bundles and that this construction goes through for every gauge natural

bundle.

We are going to describe the action in more details. We know that

S = J0 (Rm — G)/G ∼ (Rm — Tm G)0 /G ∼ J0 (Rm , G)e ∼ g — Rm— ,

1 1

=1

= =

see 17.4, and Wm G = G1

1 1

Tm G. Moreover, we have introduced the identi¬ca-

m

a¯

tion Tm G = G (g—Rm— ) with the multiplication (a, Z)(¯, Z) = (a¯, Ad(¯’1 )Z+

1

a a

¯ see 15.6. Let us now express the action of Wm G = (G1 —G) (g—Rm— ) on

1

Z), m

∼ j 1 •(0, e) ∈ W 1 G, and Y ∼ j 1 s ∈ J 1 (Rm —G),

m—

S = (g—R ). Given (A, a, Z) = =0

m 0

1

s(0) = (0, e), we have A = j0 •0 , a = pr2 —¦ •(0, e), Z = T »a’1 —¦ T0 • and ¯

1 1

Y = T0 s, where s = pr2 —¦ s. By de¬nition, Q•(j0 s) = j0 q and if we require

˜ ˜

’1

q (0) := pr2 —¦q(0) = e we have q = ρa —¦•—¦s—¦•’1 , where ρ denotes the principal

˜ 0

right action of G. Then we evaluate

’1

—¦ µ —¦ (•, s) —¦ •’1 = conj(a) —¦ µ —¦ (»a’1 —¦ •, s) —¦ •’1 .

q = ρa

˜ ¯˜ ¯˜

0 0

Hence by applying the tangent functor we get the action in form

(A, a, Z)(Y ) = Ad(a)(Y + Z) —¦ A’1 .

(1)

Proposition. For every principal bundle (P, p, M, G) the bundle of principal

connections QP is the associated ¬ber bundle W 1 P [g — Rm— , ] with the action

given by (1).

Since the standard ¬ber of QP is a Euclidean space, there are always global

sections of QP and so we have reproved in this way that every principal ¬ber

bundle admits principal connections.

17.6. The a¬ne structure on QP . In 17.2 and 17.3 we deduced that every

principal connection on P determines a bijection between principal connections

on P and the right invariant sections in C ∞ (V P — T — M ’ P ). For every

principal ¬ber bundle (P, p, M, G), let us denote by LP the associated vector

bundle P [g, Ad]. Since the fundamental ¬eld mapping (u, A) ’ ζA (u) ∈ Vu P

identi¬es V P with P — g and (ua, Ad(a’1 )(A)) ’ T Ra —¦ ζA (u), there is the

induced identi¬cation P [g, Ad] ∼ V P/G. Hence every element in LP can be

=

viewed as a right invariant vertical vector ¬eld on a ¬ber of P . Let us now

consider g — Rm— as a standard ¬ber of the vector bundle LP — T — M with the

left action of the product of Lie groups G — G1 given by

m

(a, A)(Y ) = Ad(a)(Y ) —¦ A’1 .

(1)

At the same time, we can view g — Rm— as the standard ¬ber of QP with the

1

action of Wm G given in 17.5.(1). Using the canonical a¬ne structure on the

vector space g — Rm— , we get for every two elements Y1 , Y2 ∈ g — Rm—

((A, a, Z), Y1 ) ’ ((A, a, Z), Y2 ) = Ad(a)(Y1 ’ Y2 ) —¦ A’1 ,

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

162 Chapter IV. Jets and natural bundles

cf. 15.6.(3). Hence QP is an associated a¬ne bundle to W 1 P with the modelling

vector bundle LP — T — M = W 1 P [g — Rm— ] corresponding to the action (1) of

the Lie subgroup G1 — G ‚ Wm G via the canonical homomorphism Wm G ’

1 1

m

G1 — G. Since the curvature R of a principal connection is a right invariant

m

section in C ∞ (V P — Λ2 T — M ’ P ), we can view the curvature as an operator

R : C ∞ (QP ’ M ) ’ C ∞ (LP — Λ2 T — M ’ M ). By the de¬nition, R commutes

with the action of the PBm (G)-morphisms, so that this is a typical example of

the so called gauge natural operators which will be treated in chapter XII.

17.7. Principal connections on higher order frame bundles. Let us con-

sider a frame bundle P r M and the bundle of principal connections QP r M . The

composition Q —¦ P r is a bundle functor on Mfm of order r + 1, so that there

is the canonical structure QP r M ∼ P r+1 M [gr — Rm— ], but there also is the

= m

∼ W 1 P r [gr — Rm— ; ] described in 17.6. It is an easy exer-

r

identi¬cation QP M = m

cise to verify that the former structure of an associated bundle is obtained from

the latter one by the natural reduction iM : P r+1 M ’ W 1 P r M , see proposition

15.7.

The most important case is r = 1, since the functor QP 1 associates to each

manifold M the bundle of linear connections on M . Let us deduce the coordinate

expressions of the actions of Wm G1 and G2 on (g1 — Rm— ) = Hom(Rm , gl(m)).

1

m m m

Given (A, B, Z) ∈ Wm Gm , A = (aj ) ∈ Gm , B = (bi ) ∈ G1 , Z = (zjk ) ∈

11 i 1 i

m

j

(g1 — Rm— ), “ = (“i ) ∈ (g1 — Rm— ), we have Ad(B)(Z) = (bi znj ˜n ), so that

m

bk

m m m

jk

17.5.(1) implies

(A, B, Z)(“i ) = (bi (“m + znl )˜l ˜n ).

m

ak bj

jk m nl

The coordinate expression of the homomorphism i0 : G2 ’ Wm G1 deduced in

1

m m

15.8 yields the formula

(ai , ai )(“i ) = (ai “m al an + ai al an ).

m nl ˜k ˜j nl ˜k ˜j

j jk jk

We remark that the “i introduced in this way di¬er from the classical Christo¬el

jk

symbols, [Kobayashi, Nomizu, 69], by sign and by the order of subscripts, see

17.15.

Let us mention brie¬‚y the second order case. We have to deal with (A, B, Z) ∈

Wm Gm , A = (ai ) ∈ G1 , B = (bi , bi ) ∈ G2 , Z = (zjk , zjkl ) ∈ (g2 — Rm— ). We

12 i i

m m m

j j jk

compute

Ad(B)(Z) —¦ A’1 = (bi zsm am˜s , bi zsm am˜s

p

˜k bj p p ˜l bjk

p

+ bi zmnq aq ˜n˜m + bi zmn an˜m˜s + bi zmn an˜p˜m )

p p s

˜ l b j bk ˜ l bj bk ˜ l b j bk

p ps ps

and we have to compose this action with the homomorphism i0 : G3 ’ Wm G2 .

1

m m

For every a = (ai , ai , ai ) ∈ G3 , the formula derived in 15.8 implies

m

j jk jkl

a.(“i , “i ) = ai “m al an + ai al an ,

m nl ˜k ˜j nl ˜k ˜j

jk jkl

ai “p aq an am + ai “p am as + ai “p an am as

p mnq ˜l ˜k ˜j p sm ˜l ˜jk ps mn ˜l ˜j ˜k

+ ai “s am ap an + ai aq am an + ai as am .

ps nm ˜l ˜j ˜k mnq ˜l ˜k ˜j sm ˜kj ˜l

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

17. Connections and the absolute di¬erentiation 163

17.8. The absolute di¬erential. Let us consider a ¬xed principal connection

“ : P ’ J 1 P on a principal ¬ber bundle (P, p, M, G) and an associated ¬ber

bundle E = P [S; ]. We recall the maps q : P — S ’ E and „ : P —M E ’ S, see

10.7, and we denote u : = q(u, ) : S ’ Ep(u) . Hence given local sections σ : M ’

˜

P and s : M ’ E with a common domain U ‚ M and a point x ∈ U , there is

’1

the map •σ,s : U y ’ σ(x) —¦ σ(y) —¦ s(y) ∈ Ex , i.e. •σ,s = q(σ(x), ) —¦ „ —¦ (σ, s).

In fact we use the local trivialization of E induced by σ to describe the local

behavior of s in a single ¬ber. If P and (consequently) also E are trivial bundles

and σ(x) = (0, e), then we get just the projection onto the standard ¬ber. Since

1

the principal connection “ associates to every u ∈ Px a 1-jet “(u) = jx σ of a

section σ, for every local section s : M ’ E and point x in its domain the one

jet of •σ,s at x describes the local behavior of s at x up to the ¬rst order. Our

construction does not depend on the choice of u ∈ Px , for “ is right invariant.

So we de¬ne the absolute (or covariant) di¬erential s(x) of s at x with respect

to the principal connection “ by

s(x) = jx •σ,s ∈ Jx (M, Ex )s(x) ∼ Hom(Tx M, Vs(x) E).

1 1

=

If E is an associated vector bundle, then there is the canonical identi¬cation

Vs(x) E = Ex . Then we have s(x) ∈ Hom(Tx M, Ex ) and we shall see that this

coincides with the values of the covariant derivative as de¬ned in section 11.

We can de¬ne a structure of an associated bundle on the union of the man-

1

ifolds Jx (M, Ex ), x ∈ M , where the mappings s take their values. Let

us consider the principal ¬ber bundle P 1 M —M P with the principal action

r(a1 ,a2 ) (u1 , u2 ) = (u1 .a1 , u2 .a2 ) of the Lie group G1 — G (here the dots mean

m

the obvious principal actions). We de¬ne

„ : (P 1 M —M P ) —M (∪x∈M Jx (M, Ex )) ’ Tm S

1 1

„ ((j0 f, u), jx •) = j0 (˜’1 —¦ • —¦ f ).

1 1 1

u

Let us further de¬ne a left action ¯ of G1 —G on Tm S by (remember E = P [S; ])

1

m

¯((j 1 h, a2 ), j 1 q) = j 1 ( —¦ q) —¦ j0 h’1 .

1

a2

0 0 0

One veri¬es easily that „ determines the structure of the associated bundle

E1 = (P 1 M —M P )[Tm S; ¯] and that for every section s : M ’ E its absolute

1

di¬erential s with respect to a ¬xed principal connection “ on P is a smooth

section of E1 . Hence can be viewed as an operator

: C ∞ (E) ’ C ∞ ((P 1 M —M P )[Tm S; ¯]).

1

17.9. Absolute di¬erentiation along vector ¬elds. Let E, P , “ be as in

17.8. Given a tangent vector Xx ∈ Tx M , we de¬ne the absolute di¬erentiation

in the direction Xx of a section s : M ’ E to be the value s(x)(Xx ) ∈ Vs(x) E.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

164 Chapter IV. Jets and natural bundles

Applying this procedure to a vector ¬eld X ∈ X(M ) we get a map M’

Xs:

V E with the following properties

πE —¦

(1) Xs =s

(2) s=f Xs +g s

f X+gY Y

for all vector ¬elds X, Y ∈ X(M ) and smooth functions f , g on M , πE : V E ‚

T E ’ E being the canonical projection.

So every X ∈ X(M ) determines an operator X : C ∞ (E) ’ C ∞ (V E) and

the whole procedure of the absolute di¬erentiation can be viewed as an operator

: C ∞ (T M —M E) ’ C ∞ (V E).

By the de¬nition of the connection form ¦E of the induced connection “E , it

holds

= ¦E —¦ T s —¦ X

(3) Xs

= T s —¦ X ’ (“E X) —¦ s.

(4) Xs

17.10. The frame forms. For every vector ¬eld X ∈ X(M ) and every map

s : P ’ S we de¬ne

¯

P ’ T S, T s —¦ “E X

Xs:

¯ Xs =

¯ ¯

s : P 1 M —M P ’ Tm S,

1

s(v, u) = T s —¦ T σ —¦ v,

¯ ¯ ¯

1

where “(u) = jx σ, x = p(u). We call s the absolute di¬erential of s while

¯ ¯ Xs

¯

is called the absolute di¬erential along X.

Proposition. Let s : P ’ S be the frame form of a section s : M ’ E. Then

¯

s is the frame form of s and for every X ∈ X(M ), X s is the frame form of

¯ ¯

X s.

Proof. The map X s is a section of V E = P [T S] and s(u) = „E (u, s —¦ p(u)),

¯

1 1

u ∈ P . Further, for every u ∈ Px with “(u) = jx σ, we have s(x) = jx (˜—¦¯—¦σ) ∈

us

Hom(Tx M, Vs(x) E). Hence for every X ∈ X(M ) we get X s = T u —¦ T (¯ —¦ σ) —¦ X

˜ s

and since the di¬eomorphism T S ’ (V E)x determined by u ∈ P is just T u, the

˜

frame form of X s is X s. ¯

In order to prove the other equality, let us evaluate

s(x) = {(v, u), (jx (˜’1 —¦ •)) —¦ v}.

1

u

1

Since • = u —¦ s —¦ σ, where “(u) = jx σ, the frame form of

˜¯ s is s.

¯

17.11. If E = P [S; ] is an associated vector bundle, then we can use the canon-

ical identi¬cation S ∼ Ty S for each point y ∈ S. Consider a section s : M ’ E

=

1

and its frame form s : P ’ S. Then s(x) ∈ Jx (M, Ex ) can be viewed as a

¯

value of a form Ds ∈ „¦1 (M ; E). The corresponding S-valued tensorial 1-form

D¯ : T P ’ S is de¬ned by D¯ = d¯ —¦ χ = (χ— d)(¯), where χ is the horizontal

s s s s

projection of “E . Of course, this formula de¬nes the absolute di¬erentiation

D : „¦k (P ; S) ’ „¦k+1 (P ; S) for all k ≥ 0, cf. section 11. The absolute di¬er-

entials of higher order can also be de¬ned in the nonlinear case. However, this

requires an inductive procedure and we refer the reader to [Kol´ˇ, 73 b].

ar

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

17. Connections and the absolute di¬erentiation 165

17.12. We are going to deduce a general coordinate formula for the absolute

di¬erentiation of sections of an arbitrary associated ¬ber bundle. We shall do it

in a geometric way, which reduces the problem to the proposition 16.6. For every

principal connection “ : P ’ J 1 P the image of the map “ de¬nes a reduction

“

R(“) : P 1 M —M P ’ P 1 M —M “(P ) ’ P 1 M —M J 1 P = W 1 P

’

of the principal bundle W 1 P to the structure group

G1 — G ’ G1 T m G = G1

1

(G (g — Rm— )).

m m m

˜

Let us write θ for the restriction of the canonical form θ on W 1 P to P 1 M —M

“(P ), let ω be the connection form of “ and θM will denote the canonical form

θM ∈ „¦1 (P 1 M ; Rm ).

Lemma. The following diagram is commutative

u w TP M

β— pr1

T (P 1 M —M “(P )) 1

TP

ω

u u u

θM

˜

θ

u wR

pr2 pr1

m m

R •g

g

Proof. For every u ∈ W 1 P , u = j 1 ψ(0, e), β(u) = u, we have the isomorphism

¯

u : R • g ’ Tu P and for every X ∈ Tu W P , θ(X) = u’1 (β— X). If X ∈

m 1

˜ ˜

¯

1 m

T (P M —M “(P )), we denote θ(X) = Y1 + Y2 ∈ R • g. Then u(Y1 ) = T (ψ1 —¦

˜

ψ0 )Y1 = χ(β— X) and u(Y2 ) = β— X ’ u(Y1 ) = ¦(β— X), where ¦ and χ are the

˜ ˜

vertical and horizontal projections determined by “. Since the restriction of u to

˜

m

the second factor in R •g coincides with the fundamental vector ¬eld mapping,

the commutativity of the left-hand square follows.

The commutativity of the right-hand one was proved in 16.4.

17.13. Lemma. Let s : M ’ E be a section, s : P ’ S its frame form and

¯

let s : W P ’ Tm S be the frame form of j s. Then for all u ∈ P 1 M —M P ∼

1 1 1 1

¯ =

1 1 1

P M —M “(P ) ‚ W P it holds s (u) = s(u).

¯ ¯

Proof. If u = j 1 ψ(0, e), u = β(u), then “(¯) = j 1 ψ1 (ψ0 (0)). Since we know

¯ u

s (u) = j0 („E (ψ1 —¦ ψ0 , s —¦ ψ0 )), we get s(u) = j0 (¯ —¦ ψ1 —¦ ψ0 ) = s1 (u).

1 1 1

¯ ¯ s ¯

17.14. Proposition. Let E, S, P , “, ω be as before and consider a local chart

(U, •), • = (y p ), on S. Let ei , i = 1, . . . , m be the canonical basis in Rm and e± ,

i

± = m + 1, . . . , m + dimG be a base of Lie algebra g. Let us denote θM = θM ei

the canonical form on P 1 M , ω = ω ± e± , j1 and j2 be the canonical projections

¯i

— —i ‚

on P 1 M —M P . Further, let us write ω ± = j2 ω ± , θM = j1 θM and let ·± (y) ‚yp

p

¯

be the fundamental vector ¬elds corresponding to e± . For a section s : M ’ E

let ap , ap be the coordinate functions of s on P 1 M —M P while ap be those of

¯

i

s. Then it holds

dap + ·± (aq )¯ ± = ap θM . ¯i

p

ω i

Proof. In 16.6 we described the coordinate functions bp , bp of j 1 s de¬ned on

i

W 1 P , bp = β — ap , dbp + ·± (bq )θ± = bp θi . According to 17.13, the functions ap ,

p

¯ i

ap are restrictions of bp , bp to P 1 M —M P . But then the proposition follows

i i

from lemma 17.12.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

166 Chapter IV. Jets and natural bundles

17.15. Example. We ¬nd it instructive to apply this general formula to the

simplest case of the absolute di¬erential of a vector ¬eld ξ on a manifold M

with respect to a classical linear connection “ on M . Since we consider the

standard action y i = ai y j of GL(m) on Rm , the fundamental vector ¬elds ·j oni

¯ j

Rm corresponding to the canonical basis of the Lie algebra of GL(m) are of the

‚

form δi y j ‚yk . Every local coordinates (xi ) on an open subset U ‚ M de¬ne

k

‚ ‚

a section ρ : U ’ P 1 M formed by the coordinate frames ( ‚x1 , . . . , ‚xm ) and it

holds ρ— θM = dxi . On the other hand, from the explicite equation 25.2.(2) of “

i

i

we deduce easily that the restriction of the connection form ω = (ωj ) of “ to ρ

‚

is (’“i (x)dxk ). Thus, if we consider the coordinate expression ξ i (x) ‚xi of ξ in

jk

our coordinate system and we write j ξ i for the additional coordinates of ξ,

we obtain from 17.14

‚ξ i

ξi = ’ “i ξ k .

j kj

j

‚x

Comparing with the classical formula in [Kobayashi, Nomizu, 63, p. 144], we

conclude that our quantities “i di¬er from the classical Christo¬el symbols by

jk

sign and by the order of subscripts.

Remarks

The development of the theory of natural bundles and operators is described

in the preface and in the introduction to this chapter. But let us come back

to the jet groups. As mentioned in [Reinhart, 83], it is remarkable how very

little of existing Lie group theory applies to them. The results deduced in our

exposition are mainly due to [Terng, 78] where the reader can ¬nd some more

information on the classi¬cation of Gr -modules. For the ¬rst order jet groups,

m

it is very useful to study in detail the properties of irreducible representations,

cf. section 34. But in view of 13.15 it is not interesting to extend this approach

to the higher orders. The bundle functors on the whole category Mf were ¬rst

studied by [Janyˇka, 83]. We shall continue the study of such functors in chapter

s

IX.

The basic ideas from section 15 were introduced in a slightly modi¬ed situation

by [Ehresmann, 55]. Every principal ¬ber bundle p : P ’ M with structure

group G determines the associated groupoid P P ’1 which can be de¬ned as the

factor space P — P/ ∼ with respect to the equivalence relation (u, v) ∼ (ug, vg),

u, v ∈ P , g ∈ G. Writing uv ’1 for such an equivalence class, we have two

projections a, b : P P ’1 ’ M , a(uv ’1 ) = p(v), b(uv ’1 ) = p(u). If E is a ¬ber

bundle associated with P with standard ¬ber S, then every θ = uv ’1 ∈ P P ’1

determines a di¬eomorphism qu —¦ (qv )’1 : Eaθ ’ Ebθ , where qv : S ’ Eaθ and

qu : S ’ Ebθ are the ˜frame maps™ introduced in 10.7. This de¬nes an action of

groupoid P P ’1 on ¬ber bundle E. The space P P ’1 is a prototype of a smooth

groupoid over M . In [Ehresmann, 55] the r-th prolongation ¦r of an arbitrary

smooth groupoid ¦ over M is de¬ned and every action of ¦ on a ¬ber bundle

E ’ M is extended into an action of ¦r on the r-th jet prolongation J r E

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

Remarks 167

of E ’ M . This construction was modi¬ed to the principal ¬ber bundles by

[Libermann, 71], [Virsik, 69] and [Kol´ˇ, 71b].

ar

The canonical R -valued form on the ¬rst order frame bundle P 1 M is one

m

of the basic concepts of modern di¬erential geometry. Its generalization to r-th

order frame bundles was introduced by [Kobayashi, 61]. The canonical form

on W 1 P (as well as on W r P ) was de¬ned in [Kol´ˇ, 71b] in connection with

ar

some local considerations by [Laptev, 69] and [Gheorghiev, 68]. Those canonical

forms play an important role in a generalization of the Cartan method of moving

frames, see [Kol´ˇ, 71c, 73a, 73b, 77].

ar

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

168

CHAPTER V.

FINITE ORDER THEOREMS

The purpose of this chapter is to develop a general framework for the theory

of geometric objects and operators and to reduce local geometric considerations

to ¬nite order problems. In general, the latter is a hard analytical problem and

its solution essentially depends on the category in question. Roughly speaking,

our methods are e¬cient when we deal with a su¬ciently large class of smooth

maps, but they fail e.g. for analytic maps.

We ¬rst extend the concepts and results from section 14 to a wider class of

categories. Then we present our important analytical tool, a nonlinear gener-

alization of well known Peetre theorem. In section 20 we prove the regularity

of bundle functors for a class of categories which includes Mf , Mfm , FM,

FMm , FMm,n , and we get near to the ¬niteness of the order of bundle func-

tors. It remains to deduce estimates on the possible orders of jet groups acting

on manifolds. We derive such estimates for the actions of jet groups in the cat-

egory FMm,n so that we describe all bundle functors on FMm,n . For n = 0

this reproves in a di¬erent way the classical results due to [Palais, Terng, 77]

and [Epstein, Thurston, 79] on the regularity and the ¬niteness of the order of

natural bundles.

The end of the chapter is devoted to a discussion on the order of natural

operators. Also here we essentially pro¬t from the nonlinear Peetre theorem.

First of all, its trivial consequence is that every (even not natural) local operator

depends on in¬nite jets only. So instead of natural transformations between the

in¬nite dimensional spaces of sections of the bundles in question, we have to deal

with natural transformations between the (in¬nite) jet prolongations. The full

version of Peetre theorem implies that in fact the order is ¬nite on large subsets

of the in¬nite jet spaces and, by naturality, the order is invariant under the

action of local isomorphisms on the in¬nite jets. In many concrete situations the

whole in¬nite jet prolongation happens to be the orbit of such a subset. Then all

natural operators from the bundle in question are of ¬nite order and the problem

of ¬nding a full list of them can be attacked by the methods developed in the

next chapter.

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18. Bundle functors and natural operators 169

18. Bundle functors and natural operators

Roughly speaking, the objects of a di¬erential geometric category should be

manifolds with an additional structure and the morphisms should be smooth

maps. The following approach is somewhat abstract, but this is a direct modi¬-

cation of the contemporary point of view to the concept of a concrete category,

which is de¬ned as a category over the category of sets.

18.1. De¬nition. A category over manifolds is a category C endowed with a

faithful functor m : C ’ Mf . The manifold mA is called the underlying manifold

of C-object A and A is said to be a C-object over mA.

The assumption that the functor m is faithful means that every induced map

mA,B : C(A, B) ’ C ∞ (mA, mB), A, B ∈ ObC, is injective. Taking into account

this inclusion C(A, B) ‚ C ∞ (mA, mB), we shall use the standard abuse of

language identifying every smooth map f : mA ’ mB in mA,B (C(A, B)) with a

C-morphism f : A ’ B.

The best known examples of categories over manifolds are the categories Mfm

or Mf , the categories FM, FMm , FMm,n of ¬bered manifolds, oriented man-

ifolds, symplectic manifolds, manifolds with ¬xed volume forms, Riemannian

manifolds, etc., with appropriate morphisms.

For a category over manifolds m : C ’ Mf , we can de¬ne a bundle functor on

C as a functor F : C ’ FM satisfying B —¦ F = m where B : FM ’ Mf is the

base functor. However, we have seen that the localization property of a natural

bundle over m-dimensional manifolds plays an important role. To incorporate it

into our theory, we adapt the general concept of a local category by [Eilenberg,

57] and [Ehresmann, 57] to the case of a category over manifolds.

18.2. De¬nition. A category over manifolds m : C ’ Mf is said to be local , if

every A ∈ ObC and every open subset U ‚ mA determine a C-subobject L(A, U )

of A over U , called the localization of A over U , such that

(a) L(A, mA) = A, L(L(A, U ), V ) = L(A, V ) for every A ∈ ObC and every

open subsets V ‚ U ‚ mA,

(b) (aggregation of morphisms) if (U± ), ± ∈ I, is an open cover of mA and f ∈

∞

C (mA, mB) has the property that every f —¦iU± is a C-morphism L(A, U± ) ’ B,

then f is a C-morphism A ’ B,

(c) (aggregation of objects) if (U± ), ± ∈ I, is an open cover of a manifold M

and (A± ), ± ∈ I, is a system of C-objects such that mA± = U± and L(A± , U± ©

Uβ ) = L(Aβ , U± © Uβ ) for all ±, β ∈ I, then there exists a unique C-object A

over M such that A± = L(A, U± ).

We recall that the requirement L(A, U ) is a C-subobject of A means

(i) the inclusion iU : U ’ mA is a C-morphism L(A, U ) ’ A,

(ii) if for a smooth map f : mB ’ U the composition iU —¦ f is a C-morphism

B ’ A, then f is a C-morphism B ’ L(A, U ).

There are categories like the category VB of vector bundles with no localiza-

tion of the above type, i.e. we cannot localize to an arbitrary open subset of the

total space. From our point of view it is more appropriate to consider VB (and

other similar categories) as a category over ¬bered manifolds, see 51.4.

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170 Chapter V. Finite order theorems

18.3. De¬nition. Given a local category C over manifolds, a bundle functor on

C is a functor F : C ’ FM satisfying B —¦ F = m and the localization condition:

(i) for every inclusion of an open subset iU : U ’ mA, F (L(A, U )) is the

restriction p’1 (U ) of the value pA : F A ’ mA over U and F iU is the

A

inclusion p’1 (U ) ’ F A.

A

In particular, the projections pA , A ∈ ObC, form a natural transformation

p : F ’ m. We shall see later on that for a large class of categories one can

equivalently de¬ne bundle functors as functors F : C ’ Mf endowed with such

a natural transformation and satisfying the above localization condition.

18.4. De¬nition. A locally de¬ned C-morphism of A into B is a C-morphism

f : L(A, U ) ’ L(B, V ) for some open subsets U ‚ mA, V ‚ mB. A C-object A

is said to be locally homogeneous, if for every x, y ∈ mA there exists a locally

de¬ned C-isomorphism f of A into A such that f (x) = y. The category C is called

locally homogeneous, if each C-object is locally homogeneous. A local skeleton of

a locally homogeneous category C is a system (C± ), ± ∈ I, of C-objects such that

locally every C-object A is isomorphic to a unique C± . In such a case we say

that A is an object of type ±. The set I is called the type set of C. A pointed local

skeleton of a locally homogeneous category C is a local skeleton (C± ), ± ∈ I,

with a distinguished point 0± ∈ mC± for each ± ∈ I.

A C-morphism f : A ’ B is said to be a local isomorphism, if for every

x ∈ mA there are neighborhoods U of x and V of f (x) such that the restricted

map U ’ V is a C-isomorphism L(A, U ) ’ L(B, V ). We underline that a local

isomorphism is a globally de¬ned map, which should be carefully distinguished

from a locally de¬ned isomorphism.

18.5. Examples. All the categories Mfm , Mf , FMm,n , FMm , FM are lo-

cally homogeneous. A pointed local skeleton of the category Mf is the sequence

(Rm , 0), m = 0, 1, 2, . . . , while a pointed local skeleton of the category FM is

the double sequence (Rm+n ’ Rm , 0), m, n = 0, 1, 2 . . . .

18.6. De¬nition. The space J r (A, B) of all r-jets of a C-object A into a C-

object B is the subset of the space J r (mA, mB) of all r-jets of mA into mB

generated by the locally de¬ned C-morphisms of A into B. If it is useful to

underline the category C, we write CJ r (A, B) for J r (A, B).

18.7. De¬nition. A locally homogeneous category C is called in¬nitesimally

admissible, if we have

(a) J r (A, B) is a submanifold of J r (mA, mB),

(b) the jet projections πk : J r (A, B) ’ J k (A, B), 0 ¤ k < r, are surjective

r

submersions,

(c) if X ∈ J r (A, B) is an invertible r-jet of mA into mB, then X is generated

by a locally de¬ned C-isomorphism.

Taking into account (c), we write

invJ r (A, B) = J r (A, B) © invJ r (mA, mB).

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18. Bundle functors and natural operators 171

18.8. Assume C is in¬nitesimally admissible and ¬x a pointed local skeleton

(C± , 0± ), ± ∈ I. Let us write C r (±, β) = J0± (C± , Cβ )0β for the set of all r-jets

r

of C± into Cβ with source 0± and target 0β . De¬nition 18.7 implies that every

C r (±, β) is a smooth manifold, so that the restrictions of the jet composition

C r (±, β) — C r (β, γ) ’ C r (±, γ) are smooth maps. Thus we obtain a category C r

over I called the r-th order skeleton of C.

By de¬nition 18.7, Gr := invJ0± (C± , C± )0± is a Lie group with respect to the

r

±

jet composition, which is called the r-th jet group (or the r-th di¬erential group)

of type ±. Moreover, if A is a C-object of type ±, then P r A := invJ0± (C± , A) is

r

a principal ¬ber bundle over mA with structure group Gr , which is called the

±

r-th order frame bundle of A. Let us remark that every jet group Gr is a Lie±

r

subgroup in the usual jet group Gm , m = dimC± .

For example, all objects of the category FMm,n are of the same type, so that

FMm,n determines a unique r-th jet group Gr ‚ Gr m+n in every order r. In

m,n

other words, Gm,n is the group of all r-jets at 0 ∈ Rm+n of ¬bered manifold

r

isomorphisms f : (Rm+n ’ Rm ) ’ (Rm+n ’ Rm ) satisfying f (0) = 0.

18.9. The following assumption, which deals with the local skeleton of C only,

has purely technical character.

A category C is said to have the smooth splitting property, if for every smooth

curve γ : R ’ J r (C± , Cβ ), ±, β ∈ I, there exists a smooth map “ : R — mC± ’

r

mCβ such that γ(t) = jc(t) “(t, ), where c(t) is the source of r-jet γ(t).

Since γ(t) is a curve on J r (C± , Cβ ), we know that γ(t) is generated by a

system of locally de¬ned C-morphisms. So we require that on the local skeleton

this can be done globally and in a smooth way. In all our concrete examples

the underlying manifolds of the objects of the canonical skeleton are numerical

spaces and each polynomial map determined by a jet of J r (C± , Cβ ) belongs to

C. This implies immediately that C has the smooth splitting property.

De¬nition. An in¬nitesimally admissible category C with the smooth splitting

property is called admissible.

18.10. Regularity. From now on we assume that C is an admissible category.

A family of C-morphisms f : M ’ C(A, B) parameterized by a manifold M is

said to be smoothly parameterized, if the map M —mA ’ mB, (u, x) ’ f (u)(x),

is smooth.

De¬nition. A bundle functor F : C ’ FM is called regular , if F transforms

every smoothly parameterized family of C-morphisms into a smoothly parame-

terized family of FM-morphisms.

18.11. De¬nition. A bundle functor F : C ’ FM is said to be of order r,

r ∈ N, if for any two locally de¬ned C-morphisms f and g of A into B, the

r r

equality jx f = jx g implies that the restrictions of F f and F g to the ¬ber Fx A

of F A over x ∈ mA coincide.

18.12. Associated maps. An r-th order bundle functor F de¬nes the so-called

associated maps

FA,B : J r (A, B) —mA F A ’ F B, r

(jx f, y) ’ F f (y)

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172 Chapter V. Finite order theorems

where the ¬bered product is constructed with respect to the source projection

J r (A, B) ’ mA.

Proposition. The associated maps of an r-th order bundle functor F on an

admissible category C are smooth if and only if F is regular.

Proof. By locality, it su¬ces to discuss

FC± ,Cβ : J r (C± , Cβ ) —mC± F C± ’ F Cβ .

Consider a smooth curve (γ(t), δ(t)) on J r (C± , Cβ )—mC± F C± , so that pC± δ(t) =

c(t), where c(t) is the source of r-jet γ(t). Since C has the smooth splitting

property, there exists a smooth map “ : R — mC± ’ mCβ such that γ(t) =

r

jc(t) “(t, ). The regularity of F implies µ(t) := F (“(t, ))(δ(t)) is a smooth curve

on F Cβ . By the de¬nition of the associated map, it holds FC± ,Cβ (γ(t), δ(t)) =

µ(t). Hence FC± ,Cβ transforms smooth curves into smooth curves. Now, we can

use the following theorem due to [Boman, 67]

A mapping f : Rm ’ Rn is smooth if and only if for every smooth curve

c : R ’ Rm the composition f —¦ c is smooth.

Then we conclude FC± ,Cβ is a smooth map. The other implication is obvi-

ous.

18.13. The induced action. Consider an r-th order regular bundle functor

F on an admissible category C. The ¬bers S± = F0± C± , ± ∈ I, will be called the

standard ¬bers of F . Write F±β for the restriction of FC± ,Cβ to C r (±, β) — S± ’

Sβ . In the following de¬nition we consider an arbitrary system (S± ), ± ∈ I, of

manifolds with indices from the type set of C.

De¬nition. A smooth action of C r on a system (S± ), ± ∈ I, of manifolds is a

system •±β : C r (±, β) — S± ’ Sβ of smooth maps satisfying

•βγ (b, •±β (a, s)) = •±γ (b —¦ a, s)

for all ±, β, γ ∈ I, a ∈ C r (±, β), b ∈ C r (β, γ), s ∈ S± .

By proposition 18.12, F±β are smooth maps so that they form a smooth action

of C r on the system of standard ¬bers.

18.14. Theorem. There is a canonical bijection between the regular r-th order

bundle functors on C and the smooth actions of the r-th order skeleton of C.

Proof. For every regular r-th order bundle functor F on C, F±β is a smooth

action of C r on (F0± C± ), ± ∈ I. Conversely, let (•±β ) be a smooth action of

C r on a system of manifolds (S± ), ± ∈ I. The inclusion Gr ’ C r (±, ±) gives a

±

r

smooth left action of G± on S± . For a C-object A of type ± we de¬ne GA to be

the ¬ber bundle associated to P r A with standard ¬ber S± . For a C-morphism

f : A ’ B we de¬ne Gf : GA ’ GB by

Gf ({u, s}) = {v, •±β (v ’1 —¦ jx f —¦ u, s)}

r

r r

x ∈ mA, u ∈ Px A, v ∈ Pf (x) B, s ∈ S± . One veri¬es easily that G is a well-

de¬ned regular r-th order bundle functor on C, cf. 14.22. Clearly, if we apply

the latter construction to the action F±β , we get a bundle functor naturally

equivalent to the original functor F .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

18. Bundle functors and natural operators 173

18.15. Natural transformations. Given two bundle functors F , G : C ’

FM, by a natural transformation T : F ’ G we shall mean a system of base-

preserving morphisms TA : F A ’ GA, A ∈ ObC, satisfying Gf —¦ TA = TB —¦ F f

for every C-morphism f : A ’ B. (We remark that for a large class of admissi-

ble categories every natural transformation between any two bundle functors is

formed by base-preserving morphisms, see 14.11.)

Given two smooth actions (•±β , S± ) and (ψ±β , Z± ), a C r -map

„ : (•±β , S± ) ’ (ψ±β , Z± )

is a system of smooth maps „± : S± ’ Z± , ± ∈ I, satisfying

„β (•±β (a, s)) = ψ±β (a, „± (s))

for all s ∈ S± , a ∈ C r (±, β).

Theorem. Natural transformations F ’ G between two r-th order regular

bundle functors on C are in a canonical bijection with the C r -maps between the

corresponding actions of C r .

Proof. Given T : F ’ G, we de¬ne „± : F0± C± ’ G0± C± by „± (s) = TC± (s).

One veri¬es directly that („± ) is a C r -map (F±β , F0± C± ) ’ (G±β , G0± C± ). Con-

versely, let („± ) : (•±β , S± ) ’ (ψ±β , Z± ) be a C r -map between two smooth ac-

tions of C r . Then the induced bundle functors transform A ∈ ObC of type ± into

the ¬ber bundle associated with P r A with standard ¬bers S± and Z± and we

de¬ne TA = (idP r A , „± ). One veri¬es easily that T is a natural transformation

between the induced bundle functors.

18.16. Morphism operators. We are going to generalize the concept of nat-

ural operator from 14.15 in the following three directions: 1. We replace the

category Mfm by an admissible category C over manifolds. 2. We consider the

operators de¬ned on morphisms of ¬bered manifolds. 3. We study an operator

de¬ned on some morphisms only, not on all of them. We start with the general

concept of a morphism operator.

∞

If Y1 ’ M and Y2 ’ M are two ¬bered manifolds, we denote by CM (Y1 , Y2 )

the space of all base-preserving morphisms Y1 ’ Y2 . Given another pair Z1 ’

M and Z2 ’ M of ¬bered manifolds, a morphism operator D is a map D : E ‚

∞ ∞

CM (Y1 , Y2 ) ’ CM (Z1 , Z2 ). In the case Z1 is a ¬bered manifold over Y1 , i.e. we

have a surjective submersion q : Z1 ’ Y1 , we also say that D is a base extending

operator.

In general, if we have four manifolds N1 , N2 , N3 , N4 , a map π : N3 ’ N1 and

a subset E ‚ C ∞ (N1 , N2 ), an operator A : E ’ C ∞ (N3 , N4 ) is called π-local,

if the value As(x) depends only on the germ of s at π(x) for all s ∈ E, x ∈ N3 .

k k

Such an operator is said to be of order k, 0 ¤ k ¤ ∞, if jπ(x) s1 = jπ(x) s2 implies

As1 (x) = As2 (x) for all s1 , s2 ∈ E, x ∈ N3 . We call A regular if smoothly pa-

rameterized families in E are transformed into smoothly parameterized families

in C ∞ (N3 , N4 ).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

174 Chapter V. Finite order theorems

Assume we have a surjective submersion q : Z1 ’ Y1 . Then we have de¬ned

∞ ∞

both local and k-th order operators CM (Y1 , Y2 ) ’ CM (Z1 , Z2 ) with respect to

q. Such a k-th order operator D determines the associated map

k k

D : JM (Y1 , Y2 ) —Y1 Z1 ’ Z2 , (jy s, z) ’ Ds(z),

(1) y = q(z),

∞

k

where JM (Y1 , Y2 ) means the space of all k-jets of the maps of CM (Y1 , Y2 ). If

D is regular, then D is smooth. Conversely, every smooth map (1) de¬nes a

∞ ∞

regular operator CM (Y1 , Y2 ) ’ CM (Z1 , Z2 ), s ’ D((j k s) —¦ q, ) : Z1 ’ Z2 ,

∞

s ∈ CM (Y1 , Y2 ).

18.17. Natural morphism operators. Let F1 , F2 , G1 , G2 be bundle functors

on an admissible category C. A natural operator D : (F1 , F2 ) (G1 , G2 ) is a

∞ ∞

system of regular operators DA : CmA (F1 A, F2 A) ’ CmA (G1 A, G2 A), A ∈ ObC,

∞ ∞

such that for all s1 ∈ CmA (F1 A, F2 A), s2 ∈ CmB (F1 B, F2 B) and f ∈ C(A, B)

the right-hand diagram commutes whenever the left-hand one does.

u wG A

s1 DA s1

F2 A F1 A G1 A 2

u u u u

F2 f F1 f G1 f G2 f

u wG B

s2 DB s2

F2 B F1 B G1 B 2

This implies the localization property

DL(A,U ) (s|(pF1 )’1 (U )) = (DA s)|(pG1 )’1 (U )

for every A ∈ ObC and every open subset U ‚ mA. If q : G1 ’ F1 is a natural

transformation formed by surjective submersions qA and if all operators DA are

qA -local, then we say that D is q-local.

In the special case F1 = m we have CmA (mA, F2 A) = C ∞ (F2 A), so that DA

∞

transforms sections of F2 A into base-preserving morphisms G1 A ’ G2 A; in this

(G1 , G2 ). Then D is always pG1 -local by de¬nition. If

case we write D : F2

we have a natural surjective submersion qM : G2 M ’ G1 M and we require the

(G2 ’ G1 )

values of operator D to be sections of q, we write D : (F1 , F2 )

(G2 ’ G1 ) in the special case F1 = m. In particular, if G2 is

and D : F2

of the form G2 = H —¦ G1 , where H is a bundle functor on a suitable category,

and q = pH is the bundle projection of H, we write D : (F1 , F2 ) HG1 and

D : F2 HG1 for F1 = m. In the case F1 = m = G1 , we have an operator

G2 transforming sections of F2 A into sections of G2 A for all A ∈ ObC.

D : F2

The classical natural operators from 14.15 correspond to the case C = Mfm .

Example 1. The tangent functor T is de¬ned on the whole category Mf . The

Lie bracket of vector ¬elds is a natural operator [ , ] : T • T T , see 3.10 for

the veri¬cation. Let us remark that the naturality of the bracket with respect to

local di¬eomorphisms follows directly from the fact that its de¬nition does not

depend on any coordinate construction.

Example 2. Let F be a natural bundle over m-manifolds and X be a vector

¬eld on an m-manifold M . If we apply F to the ¬‚ow of X, we obtain the ¬‚ow

of a vector ¬eld FM X on F M . This de¬nes a natural operator F : T TF.

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18. Bundle functors and natural operators 175

18.18. Natural domains. This concept re¬‚ects the situation when the oper-

ators are de¬ned on some morphisms only.

∞

De¬nition. A system of subsets EA ‚ CmA (F1 A, F2 A), A ∈ ObC, is called a

natural domain, if

(i) the restriction of every s ∈ EA to L(A, U ) belongs to EL(A,U ) for every

open subset U ‚ mA,

(ii) for every C-isomorphism f : A ’ B it holds f— (EA ) = EB , where f— (s) =

F2 f —¦ s —¦ (F1 f )’1 , s ∈ EA .

∞

If we replace CmA (F1 A, F2 A) by a natural domain EA in 18.17, we obtain the

de¬nition of a natural operator E (G1 , G2 ).

Example 1. For every admissible category m : C ’ Mf we de¬ne the C-¬elds

on the C-objects as those vector ¬elds on the underlying manifolds, the ¬‚ows

of which are formed by local C-morphisms. For every regular bundle functor

on C there is the ¬‚ow operator F : T T F de¬ned on all C-¬elds. Indeed,

if we apply F to the ¬‚ow of a C-¬eld X ∈ X(mA), we get a ¬‚ow of a vector

¬eld FX on F A. The naturality of F follows from 3.14. In particular, if C is

the category of symplectic 2m-dimensional manifolds, then the C-¬elds are the

locally Hamiltonian vector ¬elds. For the category C of Riemannian manifolds

and isometries, the C-¬elds are the Killing vector ¬elds. If C = FM, we obtain

the projectable vector ¬elds.

Example 2. The Fr¨licher-Nijenhuis bracket is a natural operator [ , ] : T —

o

Λ T • T — Λ T ’ T — Λk+l T — with respect to local di¬eomorphisms by the

k— l—

de¬nition. The functors in question do not act on the whole category Mf .

However, we have proved more than this naturality in section 8. Let us consider

∞

EM = „¦k (M ; T M ) ‚ CM (•k T M, T M ). Then we can view the bracket as

k

an operator [ , ] : (•k T • •l T, T • T ) (•k+l T, T ) with the natural domain

k l

(EM = EM —EM )M ∈ObMf and its naturality follows from 8.15. We remark that

even the Schouten-Nijenhuis bracket satis¬es such a kind of naturality, [Michor,

87b].

18.19. To deduce a result analogous to 14.17 for natural morphism operators,

we shall assume that all C-objects are of the same type and all C-morphisms

are local isomorphisms. Hence the r-th order skeleton of C is one Lie group

Gr ‚ Gr , where m is the dimension of the only object C of a local skeleton of

m

C.

Consider four bundle functors F1 , F2 , G1 , G2 on C and a q-local natural

operator D : (F1 , F2 ) (G1 , G2 ). Then the rule

k

A ’ JmA (F1 A, F2 A) —F1 A G1 A =: HA

with its canonical extension to the C-morphisms de¬nes a bundle functor H on

C. Using 18.16.(1), we deduce quite similarly to 14.15 the following assertion

Proposition. k-th order natural operators D : (F1 , F2 ) (G1 , G2 ) are in bi-

jection with the natural transformations H ’ G2 .

By 18.15, these natural transformations are in bijective correspondence with

the Gs -equivariant maps H0 ’ (G2 )0 between the standard ¬bers, where s is

the maximum of the orders of G2 and H.

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176 Chapter V. Finite order theorems

If we pose some additional natural conditions on such an operator D, they

are re¬‚ected directly in our model. For example, in the case F1 = m assume

we have a natural surjective submersion p : G2 ’ G1 and require every DA s

to be a section of pA . Then the k-order operators of this type are in bijection

with the Gs -maps f : (J k F2 )0 — (G1 )0 ’ (G2 )0 satisfying p0 —¦ f = pr2 , where

p0 : (G2 )0 ’ (G1 )0 is the map induced by p.

18.20. We are going to extend 18.19 to the case of a natural domain E ‚

(F1 , F2 ). Such a domain will be called k-admissible, if

k k

(i) the space EA ‚ JmA (F1 A, F2 A) of all k-jets of the maps from EA is a

k

¬bered submanifold of JmA (F1 A, F2 A) ’ F1 A,

k

(ii) for every smooth curve γ(t) : R ’ EC there is a smoothly parametrized

k

family st ∈ EC such that γ(t) = jc(t) st , where c(t) is the source of γ(t).

The second condition has a similar technical character as the smooth splitting

property in 18.9.

Then the rule

k

A ’ EA —F1 A G1 A =: HA

with its canonical extension to the C-morphisms de¬nes a bundle functor H on

C. Analogously to 18.19 we deduce

Proposition. If E is a k-admissible natural domain, then k-th order natural

(G1 , G2 ) are in bijection with the natural transformations H ’

operators E

G2 .

19. Peetre-like theorems

We ¬rst present the well known Peetre theorem on the ¬niteness of the order

of linear support non-increasing operators. After sketching a non-traditional

proof of this theorem, we discuss the way to its generalization and the most of

this section is occupied by the proof and corollaries of a nonlinear version of the

Peetre theorem formulated in 19.7.

19.1. Let us recall that the support supps of a section s : M ’ L of a vector

bundle L over M is the closure of the set {x ∈ M ; s(x) = 0} and for every op-

erator D : C ∞ (L1 ) ’ C ∞ (L2 ) support non-increasing means supp Ds ‚ supp s

for all sections s ∈ C ∞ (L1 ) .

Theorem, [Peetre, 60]. Consider vector bundles L1 ’ M and L2 ’ M over

the same base M and a linear support non-increasing operator D : C ∞ (L1 ) ’

C ∞ (L2 ). Then for every compact set K ‚ M there is a natural number r such

that for all sections s1 , s2 ∈ C ∞ (L1 ) and every point x ∈ K the condition

j r s1 (x) = j r s2 (x) implies Ds1 (x) = Ds2 (x).

Brie¬‚y, for any compact set K ‚ M , D is a di¬erential operator of some ¬nite

order r on K.

We shall see later that the theorem follows easily from more general results.

However the following direct (but rather sketched) proof based on lemma 19.2.

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19. Peetre-like theorems 177

contains the basic ideas of the forthcoming generalization. By the standard

compactness argument, we may restrict ourselves to M = Rm , L1 = Rm — Rn ,

L2 = Rm — Rp and to view D as a linear map D : C ∞ (Rm , Rn ) ’ C ∞ (Rm , Rp ).

19.2. Lemma. Let D : C ∞ (Rm , Rn ) ’ C ∞ (Rm , Rp ) be a support non increas-

ing linear operator. Then for every point x ∈ Rm and every real constant C > 0,

there is a neighborhood V of x and an order r ∈ N, such that for all y ∈ V \ {x},

s ∈ C ∞ (Rm , Rn ) the condition j r s(y) = 0 implies |Ds(y)| ¤ C.

Proof. Let us assume the lemma is not true for some x and C. Then we can

construct sequences sk ∈ C ∞ (Rm , Rn ) and xk ’ x, xk = x with j k sk (xk ) = 0

and |Dsk (xk )| > C and we can even require |xk ’ xj | ≥ 4|xk ’ x| for all k > j.

Further, let us choose maps qk ∈ C ∞ (Rm , Rn ) in such a way that qk (y) = 0 for

|y ’ xk | > 1 |xk ’ x|, germ sk (xk ) = germ qk (xk ), and maxy∈Rm |‚ ± qk (y)| ¤ 2’k ,

2

0 ¤ |±| ¤ k. This is possible since j k sk (xk ) = 0 for all k ∈ N and we shall not

verify this in detail. Now one can show that the map

∞

y ∈ Rm ,

q(y) := k=0 q2k (y),

is well de¬ned and smooth (note that the supports of the maps qk are disjoint). It

holds germ q(x2k ) = germ s2k (x2k ) and germ q(x2k+1 ) = 0. Since the operator D

is support non-increasing and linear, its values depend on germs only. Therefore

|Dq(x2k+1 )| = 0 and |Dq(x2k )| = |Ds2k (x2k )| > C > 0

which is a contradiction with xk ’ x and Dq ∈ C ∞ (Rm , Rp ).

Proof of theorem 19.1. Given a compact subset K we choose C = 1 and apply

lemma 19.2. We get an open cover of K by neighborhoods Vx , x ∈ K, so we can

choose a ¬nite cover Vx1 , . . . , Vxk . Let r be the maximum of the corresponding

orders. Then the condition j r s(x) = 0 implies |Ds(x)| ¤ 1 for all x ∈ K,

s ∈ C ∞ (Rm , Rn ), with a possible exception of points x1 , . . . , xk ∈ K. But if

|Ds(x)| = µ > 0, then |D( 2 s)(x)| = 2. Hence for all x ∈ K \ {x1 , . . . , xk },

µ

Ds(x) = 0 whenever j r s(x) = 0. The linearity expressed in local coordinates

implies, that this is true for the points x1 , . . . , xk as well.

If we look carefully at the proof of lemma 19.2, we see that the result does

not essentially depend on the linearity of the operator. Dealing with a nonlinear

operator, the assertion can be formulated as follows. For all sections s, q, each

point x and real constant µ > 0, there is a neighborhood V of the point x and

an order r ∈ N such that the values Dq(y) and Ds(y) do not di¬er more then

by µ for all y ∈ V \ {x} with j r q(y) = j r s(y). At the same time, there are two

essential assumptions in the proof only. First, the operator D depends on germs,

and second, the domain of D is the whole C ∞ (Rm , Rn ). Moreover, let us note

that we have used only the continuity of the values in the proof of 19.2. But the

next example shows, that having no additional assumptions on the values of the

operators, there is no reason for any ¬niteness of the order.

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178 Chapter V. Finite order theorems

19.3. Example. We de¬ne an operator D : C ∞ (R, R) ’ C 0 (R, R). For all

f ∈ C ∞ (R, R) we put

∞

dk f

’k

arctg —¦ k (x) , x ∈ R.

Df (x) = 2

dx

k=0

The value Df (x) depends essentially on j ∞ f (x).

That is why in the rest of this section we shall deal with operators with smooth

values, only. The technique used in 19.2 can be applied to more general types of

operators. We will study the π-local operators D : E ‚ C ∞ (X, Y ) ’ C ∞ (Z, W )

with a continuous map π : Z ’ X, see 18.19 for the de¬nition.

In the nonlinear case we need a general tool for extending a sequence of germs

of sections to one globally de¬ned section. In our considerations, this role will

be played by the Whitney extension theorem:

19.4. Theorem. Let K ‚ Rm be a compact set and let f± be continuous

functions de¬ned on K for all multi-indices ±, 0 ¤ |±| < ∞. There exists a

function f ∈ C ∞ (Rm ) satisfying ‚ ± f |K = f± for all ± if and only if for every

natural number m

1

’ a)β + o (|b ’ a|m )

(1) f± (b) = |β|¤m β! f±+β (a)(b

holds uniformly for |b ’ a| ’ 0, b, a ∈ K.

Let us recall that f (x) = o(|x|m ) means limx’0 f (x)x’m = 0.

The proof is rather complicated and technical and can be found in [Whitney,

34], [Malgrange, 66] or [Tougeron, 72]. If K is a one-point set, we obtain the

classical Borel theorem. We shall work with a special case of this theorem where

the compact set K consists of a convergent sequence of points in Rm . Therefore

we shall use the following assumptions on the domains of the operators.

19.5. De¬nition. A subset E ‚ C ∞ (X, Y ) is said to be Whitney-extendible, or

brie¬‚y W-extendible, if for every map f ∈ C ∞ (X, Y ), every convergent sequence

xk ’ x in X and each sequence fk ∈ E and f0 ∈ E, satisfying germ f (xk ) =

germ fk (xk ), k ∈ N, j ∞ f0 (x) = j ∞ f (x), there exists a map g ∈ E and a natural

number k0 satisfying germ g(xk ) = germ fk (xk ) for all k ≥ k0 .

19.6. Examples.

1. By de¬nition E = C ∞ (X, Y ) is Whitney-extendible.

2. Let E ‚ C ∞ (Rm , Rm ) be the subset of all local di¬eomorphisms. Then E

is W-extendible. Indeed, we need to join given germs on some neighborhood of x

only, but the original map f itself has to be a local di¬eomorphism around x, for

j ∞ f (x) = j ∞ f0 (x) and every germ of a locally de¬ned di¬eomorphism on Rm

is a germ of a globally de¬ned local di¬eomorphism. So every bundle functor F

on Mfm de¬nes a map F : E ’ C ∞ (F Rm , F Rm ) which is a pRm -local operator

with W-extendible domain.

3. Consider a ¬bered manifold p : Y ’ M . The set of all sections E = C ∞ (Y )

is W-extendible. Indeed, since we require the extension of given germs on an

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19. Peetre-like theorems 179

arbitrary neighborhood of the limit point x only, we may restrict ourselves to a

local chart Rm — Rn ’ Y . Now, we can work with the coordinate expressions

of the given germs of sections, i.e. with germs of functions. The existence of the

˜extension™ f of given germs implies that the germs of coordinate functions satisfy

condition 19.4.(1), and so there are functions joining these germs. But these

functions represent a coordinate expression of the required section. Therefore

the operators dealt with in 19.1 are idM -local linear operators with W-extendible

domains.

19.7. Nonlinear Peetre theorem. Now we can formulate the main result of

this section. The last technical assumption is that for our π-local operators, the

map π should be locally non-constant, i.e. there are at least two di¬erent points

in the image π(U ) of any open set U .

Theorem. Let π : Z ’ Rm be a locally non-constant continuous map and

let D : E ‚ C ∞ (Rm , Rn ) ’ C ∞ (Z, W ) be a π-local operator with a Whitney-

extendible domain. Then for every ¬xed map f ∈ E and for every compact subset

K ‚ Z there exist a natural number r and a smooth function µ : π(K) ’ R which

is strictly positive, with a possible exception of a ¬nite set of points in π(K),

such that the following statement holds.

For every point z ∈ K and all maps g1 , g2 ∈ E satisfying |‚ ± (gi ’ f )(π(z))| ¤

µ(π(z)), i = 1, 2, 0 ¤ |±| ¤ r, the condition

j r g1 (π(z)) = j r g2 (π(z))

implies

Dg1 (z) = Dg2 (z).

Before going into details of the proof, we present some remarks and corollaries.

19.8. Corollary. Let X, Y , Z, W be manifolds, π : Z ’ X a locally non-

constant continuous map and let D : E ‚ C ∞ (X, Y ) ’ C ∞ (Z, W ) be a π-local

operator with Whitney-extendible domain. Then for every ¬xed map f ∈ E and

for every compact set K ‚ Z, there exists r ∈ N such that for every x ∈ π(K),

g ∈ E the condition j r f (x) = j r g(x) implies

Df |(π ’1 (x) © K) = Dg|(π ’1 (x) © K).

19.9. Multilinear version of Peetre theorem. Let us note that the classical

Peetre theorem 19.1 follows easily from 19.8. Indeed, idM -locality is equivalent to

the condition on supports in 19.1, the sections of a ¬bration form a W-extendible

domain (see 19.6), so we can apply 19.8 to the zero section of the vector bundle

L1 ’ M . Hence for every compact set K ‚ M there is an order r ∈ N such that

Ds(x) = 0 whenever j r s(x) = 0, x ∈ K, s ∈ C ∞ (L), and the classical Peetre

theorem follows.

But applying the full formulation of theorem 19.7, we can prove in a similar

way a ˜multilinear base-extending™ Peetre theorem.

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180 Chapter V. Finite order theorems

Theorem. Let L1 , . . . , Lk be vector bundles over the same base M , L ’ N

be another vector bundle and let π : N ’ M be continuous and locally non-

constant. If D : C ∞ (L1 ) — · · · — C ∞ (Lk ) ’ C ∞ (L) is a k-linear π-local operator,

then for every compact set K ‚ N there is a natural number r such that for every

x ∈ π(K) and all sections s, q ∈ C ∞ (L1 •· · ·•Lk ) the condition j r s(x) = j r q(x)

implies

Ds|(π ’1 (x) © K) = Dq|(π ’1 (x) © K).

Proof. We may assume Li = Rm — Rni , i = 1, . . . , k. Then all assumptions

of 19.7 are satis¬ed and so, chosen a compact set K ‚ N and the zero section

of L1 • · · · • Lk , we get some order r and a function µ : π(K) ’ R. Consider

arbitrary sections q, s ∈ C ∞ (L1 • · · · • Lk ) and a point x ∈ π(K), µ(x) > 0.

Using multiplication of sections by positive real constants, we can arrange that

all their derivatives up to order r at the point x are less then µ(x). Hence if

j r q(x) = j r s(x), then for a suitable c > 0, c ∈ R, it holds

ck · Ds(z) = D(c · s)(z) = D(c · q)(z) = ck · Dq(z)

for all z ∈ K © π ’1 (x). According to 19.7, the function µ can be chosen in such

a way that the set {x ∈ π(K); µ(x) = 0} is discrete. So the theorem follows from

the multilinearity of the operator and the continuity of its values, what is easily

checked looking at the coordinate description of the multilinear operators.

19.10. One could certainly replace the Whitney extendibility by some other

property, but this cannot be completely omitted. To see this, consider the opera-

tor constructed in 19.3 and let us restrict its domain to the subset E ‚ C ∞ (R, R)

of all polynomials. We get an operator D : E ’ C ∞ (R, R) essentially depending

on in¬nite jets. Also the requirement on π is essential because dropping it, any

action of the group of germs of maps f : (Rm , 0) ’ (Rm , 0) on a manifold should

factorize to an action of some jet group Gr . m

Let us notice that the assertion of our theorem is near to local ¬niteness of the

order with respect to the topology on Z and to the compact open C ∞ -topology

on C ∞ (Rm , Rn ), see e.g. [Hirsch, 76] for de¬nition. It would be su¬cient if we

might always choose a strictly positive function µ : π(K) ’ R in the conclusion

of the theorem. However, example 19.15 shows that this need not be possible in

general. On the other hand, if we add a suitable regularity condition, then the

mentioned local ¬niteness can be proved. Regularity will mean that smoothly

parameterized families of maps in the domain are transformed into smoothly

˜

parameterized families. The idea of the proof is to de¬ne a new operator D

˜

with domain E formed by all one-parameter families of maps, then to perform

˜

a similar construction as in the proof of 19.7 and to apply theorem 19.7 to D to

get a contradiction, see [Slov´k, 88]. Therefore, beside the regularity, we need

a

˜ is also W-extendible. This is not obvious in general, but it is evident if

that E

E consists of all sections of a ¬bration. Since we shall mostly deal with regular

operators de¬ned on all sections of a ¬bration, we present the full formulation.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

19. Peetre-like theorems 181

Theorem. Let Z, W be manifolds, Y ’ X a ¬bration, π : Z ’ X a locally

non-constant map and let D : E = C ∞ (Y ’ X) ’ C ∞ (Z, W ) be a regular

π-local operator. Then for every ¬xed map f ∈ E and for every compact set

K ‚ Z, there exist an order r ∈ N and a neighborhood V of f in the compact

open C ∞ -topology such that for every x ∈ π(K) and all g1 , g2 ∈ V © E the

condition

j r g1 (x) = j r g2 (x)

implies

Dg1 |(π ’1 (x) © K) = Dg2 |(π ’1 (x) © K).

Similar, but essentially weaker, results can also be deduced dealing with op-

erators with continuous values, see [Chrastina, 87], [Slov´k, 87 b].

a

Let us pass to the proof of 19.7. In the sequel, we ¬x manifolds Z, W , a

locally non-constant continuous map π : Z ’ Rm , a Whitney-extendible subset

E ‚ C ∞ (Rm , Rn ) and a π-local operator D : E ’ C ∞ (Z, W ). The proof is

based on two lemmas.

19.11. Lemma. Let z0 ∈ Z be a point, x0 := π(z0 ), f ∈ E, and let us de¬ne

a function µ : Rm ’ R by µ(x) = exp(’|x ’ x0 |’1 ) if x = x0 and µ(x0 ) = 0.

Then there is a neighborhood V of the point z0 ∈ Z and a natural number

r such that for every z ∈ V ’ π ’1 (x0 ) and all maps g1 , g2 ∈ E satisfying

|‚ ± (gi ’ f )(π(z))| ¤ µ(π(z)), i = 1,2, 0 ¤ |±| ¤ r, the condition j r g1 (π(z)) =

j r g2 (π(z)) implies Dg1 (z) = Dg2 (z).

Proof. We assume the lemma does not hold and we shall ¬nd a contradiction.

If the assertion is not true, then we can construct sequences zk ’ z0 in Z,

xk := π(zk ) ’ x0 and maps fk , gk ∈ E satisfying for all k ∈ N

|‚ ± (fk ’ f )(xk )| ¤ µ(xk ) for all 0 ¤ |±| ¤ k

(1)

j k fk (xk ) = j k gk (xk )

(2)

(3) Dfk (zk ) = Dgk (zk ).

Since all xk are di¬erent from x0 , by passing to subsequences we can assume

1

|xk+1 ’ x0 | < |xk ’ x0 |.

(4)

4

Let us ¬x Riemannian metrics ρZ or ρW on Z or W , respectively, and choose

further points zk ∈ Z, zk ’ z0 , xk := π(¯k ) and neighborhoods Uk or Vk of xk

¯ ¯ ¯ z

or xk , respectively, in such a way that for all k ∈ N the following six conditions

¯

hold

|xk ’ x0 | ¤ 2|a ’ b| for all a ∈ Uk ∪ Vk , b ∈ Uj ∪ Vj , k = j

(5)

|‚ ± (fk ’ f )(a)| ¤ 2µ(xk ) for all a ∈ Uk ∪ Vk , 0 ¤ |±| ¤ k

(6)

|‚ ± (gk ’ f )(a)| ¤ 2µ(xk ) for all a ∈ Uk ∪ Vk , 0 ¤ |±| ¤ k

(7)

ρW (Dgk (zk ), Dfk (¯k )) ≥ kρZ (zk , zk )

(8) z ¯

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182 Chapter V. Finite order theorems

and for all m, k ∈ N, and multi-indices ± with |±| + 2m ¤ k, a ∈ Uk , and b ∈ Vk

we require

1 1 ±+β 1

fk (b)(a ’ b)β ’ ‚ ± gk (a) ¤

(9) ‚

|b ’ a|m β! k

|β|¤m

1 1 ±+β 1

gk (a)(b ’ a)β ’ ‚ ± fk (b) ¤ .

(10) ‚

|b ’ a|m β! k

|β|¤m

All these requirements can be satis¬ed. Indeed, the equalities (5), (6), (7) are

valid for all points a, b from some suitable neighborhoods Wk of the points xk .

By the Taylor formula, for any ¬xed k and |±| + m ¤ k, (2) implies ‚ ± gk (a) =

‚ ± fk (a) + o(|a ’ xk |m ). Therefore, if we consider only points a, b ∈ Wk such

that

|b ’ xk | ¤ 2|b ’ a|, |a ’ xk | ¤ 2|b ’ a|,

(11)

then under the condition |±| + 2m ¤ k we get ( note that o(|a ’ xk |m ) or

o(|b ’ xk |m ) now implies o(|a ’ b|m ))

1 ±+β 1 ±+β

fk (b)(a ’ b)β = gk (b)(a ’ b)β + o(|a ’ b|m )

‚ ‚

β! β!

|β|¤m |β|¤m

‚ ± gk (a) + o(|b ’ a|m )

=

1 ±+β 1 ±+β

gk (a)(b ’ a)β = fk (a)(b ’ a)β + o(|a ’ b|m )

‚ ‚

β! β!

|β|¤m |β|¤m

‚ ± fk (b) + o(|b ’ a|m ).

=

Hence also conditions (9), (10) are realizable if we take Uk , Vk in su¬ciently small

neighborhoods Wk of xk in such a way that (11) holds for all a ∈ Uk , b ∈ Vk . By

virtue of (3), there are also neighborhoods of the points zk in Z ensuring (8).

Finally, we are able to choose appropriate points zk and neighborhoods Uk , Vk

¯

using the fact that π is continuous and locally non-constant.

The aim of conditions (1), (4)“(7), (9), (10) is to guarantee the existence of

a map h ∈ C ∞ (Rm , Rn ) satisfying

(12) germ h(xk ) = germ gk (xk ) and germ h(¯k ) = germ fk (¯k ).

x x

Then, by virtue of our requirements on E, we may assume h ∈ E, provided we

use (12) for large indices k, only. But applying D to h, the π-locality and (8)

imply

ρW (Dh(zk ), Dh(¯k )) ≥ kρZ (zk , zk )

z ¯

for large k™s, and this is a contradiction with Dh ∈ C ∞ (Z, W ) and (zk , zk ) ’

¯

(z0 , z0 ).

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19. Peetre-like theorems 183

So it remains to verify condition 19.4.(1) in the Whitney extension theorem

¯ ¯ ¯

with K = k (Uk ∪ Vk ) ∪ {x0 } and f± (x) = ‚ ± gk (x) if x ∈ Uk , f± (x) = ‚ ± fk (x)

¯

if x ∈ Vk and f± (x0 ) = ‚ ± f (x0 ). This follows by our construction for all couples

¯ ¯

(a, b) ∈ k (Uk — Vk ), see (9), (10). In all other cases and for all m ∈ N we have